TRACTOR DYNAMICS AS A COMPLEX MECHANICAL SYSTEM SPATIAL FRAME TYPE

Purpose of the study is to develop a matrix method for studying the dynamics of a tractor as a multi-mass spatial system of rigid bodies with an arbitrary arrangement of elastic suspension of bodies on shock absorbers relative to a fixed support surface and the presence of elastic connections between the bodies, made in the form of beam elements. Research methods. The methodological basis of the work is the generalization and analysis of well-known scientific results regarding the dynamics of two-mass systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, determine the goal and formulate the research objectives. When creating empirical models, the main provisions of the theory of stability of systems, methodology of systems analysis and research of operations were used. The results of the study. A wheeled vehicle is presented as an amortized continuous frame type structure with assemblies and assembly units located on it, as well as a methodology for calculating individual block matrices of stiffness and damping coefficients. In this case, it is assumed that a viscous damper can be connected in parallel to each elastic element. In this construction of the stiffness and damping matrix of the block matrix are formed in the same way. Damping matrices are derived from the corresponding matrices by substituting damping constants instead of stiffness constants. To determine the natural frequencies and vibration modes of an undamped system using a PC, the most effective method of diagonalization by successive rotations. This method provides a complete solution to the problem, allowing all frequencies and shapes to be determined simultaneously, and good convergence. Conclusions. The considered method for analyzing and calculating the dynamics and vibration damping of a tractor as a complex mechanical system is based on a matrix record of the problem of spatial vibrations of a system of rigid bodies with elastic bonds. Matrix equations seem to be especially useful in the study of complex tightly coupled systems with the obligatory use of a PC. The presented work provides a complete methodology for calculating a tractor as a complex mechanical system such as a spatial frame with equipment installed on it.

Evaluation of indicators of curvilinear movement of a road train using mathematical simulation

Introduction: a widespread approach to the transportation of large-sized and heavy-weight indi-visible cargo on roads and terrain is the use of multi-axle wheeled transport complexes, which are road trains. At the same time, due to the significant overall dimensions, one of the most important properties of such machines is agility, that is, the ability to move along a trajectory of large curvature in a limited area, which is especially important in loading / unloading zones. Subject of research: the article presents an approach to predicting the indicators of curvilinear movement of multi-axle wheeled road trains, based on the application of the method of mathemati-cal modeling of the dynamics of body systems. Methodology and methods: the essence of the method is to create a mathematical model of the movement of a road train, represented by a system of rigid bodies, which are interconnected by kin-ematic and power connections. The simulation model developed within the framework of the study makes it possible to take into account with high accuracy the peculiarities of the interaction of the wheel propeller with the supporting surface, the redistribution of normal reactions between the sup-port modules, as well as the force factors arising in the coupling device and ensuring the interaction between the tractive vehicle and the trailer link. The mathematical description of the interaction of the propeller with the ground is based on the concept of “friction ellipse”. Using the presented mod-el, an assessment of the turnability of a wheeled road train equipped with a trailed link with swivel and non-swivel wheels was carried out. The required width along the tracks of the outer (running in) and inner (lagging) wheels was used as an assessment criterion. To assess the feasibility of using a trailed link with fully steered wheels and, accordingly, complicating the design of the machine, an additional assessment of the required power of the steering drive was carried out. Results and scientific novelty: a mathematical model of the dynamics of a road train was de-veloped. It makes possible to predict with high accuracy the indicators of curvilinear movement of wheeled vehicles, as well as to estimate the required power of the steering drive. Practical significance: a mathematical model of road train movement was developed, which al-lows a wide range of tests to be carried out to assess not only the indicators of curved-linear move-ment, but also the mobility of wheeled vehicles of any configuration as a whole.

Rocking Mechanism of Masonry Portals with Circular Arches

Unreinforced masonry (URM) structures represent most of the world architectural heritage, whose vulnerability has been also highlighted by damages and collapses occurred after recent seismic events. Numerous studies regarding the seismic capacity of masonry walls, arches and portals have been carried out by applying the so-called equivalent static analysis method, neglecting their dynamic behaviour. A proper evaluation of the dynamic response of masonry elements can be done analytically considering the dynamic equation of rigid bodies not resistant to the tensile stresses. Some studies are available in literature regarding the dynamic behaviour of walls and arches. In this framework, the paper aims to develop an analytical model, able to describe the dynamic behaviour of portals with circular arches, subjected to a base motion. Starting point of the analysis is the evaluation of the mechanism (local, semi-global or global) governing the activation of the motion of the structure, performed in the context of Limit Analysis. Subsequently the equation of motion of the system of rigid bodies is derived applying the Lagrange Equation. Finally a numerical application is carried out.

Cinemática 2D de caballos al trote mediante videometría y modelamiento matemático

Currently, the direct observation method is used to assess the movement of horses. However, this method is limited, totally subjective and many details of the functionality of the musculoskeletal system cannot be detected and evaluated, because they are not perceptible to the naked eye.The aim of this study was to develop a mathematical model that calculates, plots and simulates the 2D angular movement of some horse joints. The horse's locomotor system was modeled as a mechanical system of rigid bodies articulated by 15 simple joints.The mathematical solution of the mechanism was made from the standpoint of liabilities of inverse kinematics (flat), 15 link equations were constructed associating the body segments of the system in movement with the inertial base and the mathematical optimization method based on the least squares calculation was used.The kinematic curves of the main joints were obtained in this study; trajectories (height) of the markers on fore and hind coronary band (hoofs), and a simulation of the mechanical system. This tool removes the subjectivity and enables veterinarians to observe, evaluate (qualitatively and quantitatively), diagnose and research about different phenomena of the horse’s gait.

Energetically Consistent Calculations for Oblique Impact in Unbalanced Systems With Friction

Analytical mechanics is used to derive original 3D equations of motion that represent impact at a point in a system of rigid bodies. For oblique impact between rough bodies in an eccentric (unbalanced) configuration, these equations are used to compare the calculations of energy dissipation obtained using either the kinematic, the kinetic, or the energetic coefficient of restitution (COR); eN,eP, or e*. Examples demonstrate that for equal energy dissipation by nonfrictional sources, either eN≤e*≤eP or eP≤e*≤eN depending on whether the unbalance of the impact configuration is positive or negative relative to the initial direction of slip. Consequently, when friction brings initial slip to rest during the contact period, calculations that show energy gains from impact can result from either the kinematic or the kinetic COR. On the other hand, the energetic COR always correctly accounts for energy dissipation due to both hysteresis of the normal contact force and friction, i.e., it is energetically consistent.

Improving Techniques in Statically Equivalent Serial Chain Modeling for Center of Mass Estimation

Any articulated system of rigid bodies defines a statically equivalent serial chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. An SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed, followed by the experimental validation of SESC modeling. Second, the problem of generating an SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.